Has any US president, during their tenure in office, accidentally told the public or the press secret or top secret information?
Bonus question: Is there an official procedure for if this were to happen?
Yes, Several times in the lead up to the war with Iraq, George W Bush revealed information from his PDB that was not only secret but his revealing of the information put some of our agents at risk of exposure.
However the president is the ultimate classification authority, meaning that the person holding the office of POTUS, can classify or declassify government documents as the needs of the country dictate. Unlike when aides and staffers accidentally/incidentally reveal classified\secret information there is no reporting of the breach required.
Jimmy Carter in an open forum after he killed the B1 bomber told the public we were developing stealth aircraft years before we had them.He was trying to win an election and broke the news to establish his defense creds.